Crazy Krypton

The mass of a planet creates gravity. If two planets are the same size and one has twice the mass as the other than it will also have a gravitational acceleration that is twice as big. On earth the acceleration of gravity is g=9.8ish m/s^2. For guesstimating 9.8 is rounded to a nice g=10 m/s^2.

How high is a tall building? 1 story? Nope. 10 stories. No. 100 stories. Yes 100 stories (the Chrystler Building is 77 stories and the Empire State Building is 102 stories. How high is each story? 9ft.

So that means that the total height of a tall building is 900ft. To make ft into meters we multiply by 0.3 making a tall building 300m tall.

Relative to the ground, the potential energy at the top of the building is PE = m*g*h. All we need to do is guess the mass of Superman. Let's use a nice round 100kg (He's not 10 kg and he's definitely not 1000kg (that's a ton) so the geometric mean is 100kg). So at the top of his jump Supes' potential energy is 100kg * 10m/s^2 * 300m = 3 * 10^5 J (J=joules which we can think of as shorthand for all those more basic units which went into the calculation or we can just leave them whatever).

So then in order to get that high he must start out with a kinetic energy which is that big: KE=PE. 1 / 2 m*v^2 = 3*10^5kg m^2 / s^2. We said Superman's mass was 100kg so the velocity is v=sqrt(6*10^3). The square root of 10^3 is 30 because we divide the exponent 3 in half and multiply the coefficient by 3 for the 0.5 that is left in the exponent. The square root of 6? I don't know but I know that the square root of 4 is 2 and the square root of 9 is 3 and 6 is between 4 and 9 so I'll say 2.5.

So Superman's speed when he jumps must be v=2.5 * 30 = 75m/s. That's pretty fast but not nearly as fast as sound (340m/s) so he doesn't have to worry about causing sonic booms. How about as fast as a bullet? Not a chance Check out the Orders of Magnitude wikipedia page. A bullet moves at like 3 times the speed of sound or 1000 m/s.

So if he jumps at 75m/s, what kind of force must his legs exert to get that fast? F = m * a and he starts from standing still and gets to 75m/s. The acceleration is a = v / t. We need some guess at how long he jumps for.

The golden age Superman wasn't really quicker than the average person. He could run and move fast but he sort of jumped normal so let's estimate how long a person jumps for before their feet leave the ground. 1 second? No way. That's a pretty long time. 0.1 second? That seems a little short. Let's take the geometric mean and say 0.3 seconds. Then the acceleration is a = 75m/s / 0.3s = 250m/s^2. And then the force his legs exert is F=ma=100kg*250m/s^2 = 2.5*10^4N (N is newton's which is the unit for force or weight).

That seems big but what force does the average person exert? 1ft? Ya... It seems small but 1ft * 3 = 0.3m and an average person definitely can't jump 1m so let's just pick a number in between 50cm seems like a good guess. So the average person's potential energy at the top of their jump is PE = mgh= 100kg * 10m/s * 0.5m = 500J which means they need an initial velocity:
KE = 0.5 mv^2 = 500J ---> v^2 = 10m^2/s^2 ---> v=3m/s.

So the average person's acceleration is a = v/t = 3 / 0.3 = 10m/s^2. And the total force they jumped with is then F = ma = 100*10 = 1000N = 10^3N.

Superman's jump was 2.5 * 10^4N and and average person's was 10^3N. So Superman's jump is 25 times stronger than an average person's.

So to make Superman jump like a normal person on Krypton, the gravity must be 25 times greater. The gravitational acceleration on Krypton would then be 25 * 10m/s^2 = 250m/s^2. To get a gravitational acceleration that is 25 times greater on a planet the same size as earth, the mass of Krypton would have had to have been 25 times greater too. The mass of Earth is about 6*10^24kg which means the mass of Krypton would have been 1.5*10^26.

You should really read Physics of Superheroes to see what this means about what must have caused Krypton's destruction.

On that note, I think the Fermi Problem I pose tomorrow will be about Krypton's explosion.